The graph of $y=x^x$ looks like this:
As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$.
I know that $e$ is the number for exponential growth and $\frac{d}{dx}e^x=e^x$, but these ideas seem unrelated to the fact that the mimum value of $x^x$ is $1/e$. Is this just pure coincidence, or could someone provide an intuitive explanation (i.e. more than just a proof) of why this is?